Many problems can be expressed in terms of a numeric constraint satisfaction problem over finite or continuous domains (numeric CSP). The purpose of this paper is to show that the consistency techniques that have been developed for CSPs can be adapted to numeric CSPs. Since the numeric domains are ordered the underlying idea is to handle domains only by their bounds. The semantics that have been elaborated, plus the complexity analysis and good experimental results, confirm that these techniques can be used in real applications. 1

This paper discusses how interval analysis can be used to solve a wide variety of problems in computer graphics. These problems include ray tracing, interference detection, polygonal decomposition of parametric surfaces, and CSG on solids bounded by parametric surfaces. Only two basic algorithms are required: SOLVE, which computes solutions to a system of constraints, and MINIMIZE, which computes the global minimum of a function, subject to a system of constraints. We present algorithms for SOLVE and MINIMIZE using interval analysis as the conceptual framework. Crucial to the technique is the creation of "inclusion functions" for each constraint and function to be minimized. Inclusion functions compute a bound on the range of a function, given a similar bound on its domain, allowing a branch and bound approach to constraint solution and constrained minimization. Inclusion functions also allow the MINIMIZE algorithm to compute global rather than local minima, unlike many other numerica...

The design and implementation of constraint logic programming (CLP) languages over intervals is revisited. Instead of decomposing complex constraints in terms of simple primitive constraints as in CLP(BNR), complex constraints are manipulated as a whole, enabling more sophisticated narrowing procedures to be applied in the solver. This idea is embodied in a new CLP language Newton whose operational semantics is based on the notion of box-consistency, an approximation of arc-consistency, and whose implementation uses Newton interval method. Experimental results indicate that Newton outperforms existing languages by an order of magnitude and is competitive with some state-of-the-art tools on some standard benchmarks. Limitations of our current implementation and directions for further work are also identified.

This paper presents Newton, a branch & prune algorithm to find all isolated solutions of a system of polynomial constraints. Newton can be characterized as a global search method which uses intervals for numerical correctness and for pruning the search space early. The pruning in Newton consists in enforcing at each node of the search tree a unique local consistency condition, called box-consistency, which approximates the notion of arc-consistency well-known in artificial intelligence. Box-consistency is parametrized by an interval extension of the constraint and can be instantiated to produce the Hansen-Segupta's narrowing operator (used in interval methods) as well as new operators which are more effective when the computation is far from a solution. Newton has been evaluated on a variety of benchmarks from kinematics, chemistry, combustion, economics, and mechanics. On these benchmarks, it outperforms the interval methods we are aware of and compares well with state-of-the-art continuation methods. Limitations of Newton (e.g., a sensitivity to the size of the initial intervals on some problems) are also discussed. Of particular interest is the mathematical and programming simplicity of the method.

Incomplete knowledge of the structure of mechanisms is an important fact of life in reasoning, commonsense or expert, about the physical world. Qualitative simulation captures an important kind of incomplete, ordinal, knowledge, and predicts the set of qualitatively possible behaviors of a mechanism, given a qualitative description of its structure and initial state. However, one frequently has quantitative knowledge as well as qualitative, though seldom enough to specify a numerical simulation.

We start with a mathematical definition of a real interval as a closed, connected set of reals. Interval arithmetic operations (addition, subtraction, multiplication and division) are likewise defined mathematically and we provide algorithms for computing these operations assuming exact real arithmetic. Next, we define interval arithmetic operations on intervals with IEEE 754 floating point endpoints to be sound and optimal approximations of the real interval operations and we show that the IEEE standard’s specification of operations involving the signed infinities, signed zeros, and the exact/inexact flag are such as to make a correct and optimal implementation more efficient. From the resulting theorems we derive data that are sufficiently detailed to convert directly to a program for efficiently implementing the interval operations. Finally we extend these results to the case of general intervals, which are defined as connected sets of reals that are not necessarily closed. 1

We present an efficient and robust algorithm for finding points of collision between time-dependent parametric and implicit surfaces. The algorithm detects simultaneous collisions at multiple points of contact. When the regions of contact form curves or surfaces, it returns a finite set of points uniformly distributed over each contact region. Collisions can be computed for a very general class of surfaces: those for which inclusion functions can be constructed. Included in this set are the familiar kinds of surfaces and time behaviors encountered in computer graphics. We use a new interval approach for constrained minimization to detect collisions, and a tangency condition to reduce the dimensionality of the search space. These approaches make interval methods practical for multi-point collisions between complex surfaces. An interval Newton method based on the solution of the interval linear equation is used to speed convergence to the collision time and location. This method is mor...

. Compared to standard numerical methods for initial value problems (IVPs) for ordinary differential equations (ODEs), validated methods for IVPs for ODEs have two important advantages: if they return a solution to a problem, then (1) the problem is guaranteed to have a unique solution, and (2) an enclosure of the true solution is produced. The authors survey Taylor series methods for validated solutions of IVPs for ODEs, describe several such methods in a common framework, and identify areas for future research. Key words. initial value problems, ordinary differential equations, interval arithmetic, Taylor series methods. AMS subject classifications. 65L05, 65G10, 65L60. 1. Introduction. We consider validated numerical methods for the solution of the autonomous initial-value problems (IVPs) y 0 (t) = f(y); y(t 0 ) = y 0 ; (1.1) where t 2 [t 0 ; T ] for some T ? t 0 . Here t 0 ; T 2 R,f 2 C k\Gamma1 (D), D ` R n is an open set, f : D ! R n , and y 0 2 D. For expositional c...

. The role of the interval subdivision selection rule is investigated in branch-and-bound algorithms for global optimization. The class of rules that allow convergence for the model algorithm is characterized, and it is shown that the four rules investigated satisfy the conditions of convergence. A numerical study with a wide spectrum of test problems indicates that there are substantial differences between the rules in terms of the required CPU time, the number of function and derivative evaluations and space complexity, and two rules can provide substantial improvements in efficiency. Key words. global optimization, interval arithmetic, interval subdivision AMS subject classifications. 65K05, 90C30 Abbreviated title: Subdivision directions in interval methods. 1. Introduction. Interval subdivision methods for global optimization [7, 21] aim at providing reliable solutions to global optimization problems min x2X f(x) (1) where the objective function f : IR n ! IR is continuo...

. Interval Newton methods in conjunction with generalized bisection can form the basis of algorithms that find all real roots within a specified box X ae R n of a system of nonlinear equations F (X) = 0 with mathematical certainty, even in finite-precision arithmetic. In such methods, the system F (X) = 0 is transformed into a linear interval system 0 = F (M) +F 0 (X)( ~ X \Gamma M); if interval arithmetic is then used to bound the solutions of this system, the resulting box ~ X contains all roots of the nonlinear system. We may use the interval Gauss--Seidel method to find these solution bounds. In order to increase the overall efficiency of the interval Newton / generalized bisection algorithm, the linear interval system is multiplied by a preconditioner matrix Y before the interval Gauss--Seidel method is applied. Here, we review results we have obtained over the past few years concerning computation of such preconditioners. We emphasize importance and connecting relationships,...